Statistical & Econometric
Yule’s Q & Yule’s Y Coefficient of Colligation for 2×2 Contingency Tables
Normalize 2×2 odds ratios onto [-1, +1] marginal-invariant scales using George Udny Yule’s association Q = (ad - bc) / (ad + bc) and colligation Y = (SQRT(ad) - SQRT(bc)) / (SQRT(ad) + SQRT(bc)).
Why Pearson φ shrinks when disease prevalence changes—while Yule’s Q and Y stay invariant!
In 2×2 epidemiological, diagnostic, and NLP co-occurrence tables (cells a, b, c, d), the phi coefficient φ cannot reach +1.0 unless row and column marginal prevalence rates match! George Udny Yule (1900, 1912) mapped the cross-product odds ratio OR = (a * d) / (b * c) onto symmetric [-1, +1] bounds that remain 100% invariant to row or column rescaling: Yule’s Q = (OR - 1) / (OR + 1) = (a*d - b*c) / (a*d + b*c) (identical to Goodman-Kruskal γ for 2×2 tables) and Yule’s Coefficient of Colligation Y = (SQRT(OR) - 1) / (SQRT(OR) + 1) = (SQRT(a*d) - SQRT(b*c)) / (SQRT(a*d) + SQRT(b*c))!
For a 2×2 biomarker cohort with a = 42 true positives, b = 14 false positives, c = 12 false negatives, and d = 52 true negatives (OR = 2,184 / 168 = 13.00), Yule’s Q = +0.8571 and Yule’s conservative colligation Y = +0.5657!
| 2×2 Contingency Table (a, b, c, d) | Odds Ratio ad / bc | Yule’s Q (-1 to +1) | Yule’s Colligation Y (-1 to +1) |
|---|---|---|---|
| Diagnostic Cohort (42, 14, 12, 52) | 13.00 | +0.8571 (Strong Positive) | +0.5657 |
| Statistical Independence (30, 30, 20, 20) | 1.00 | 0.0000 (Null Association) | 0.0000 |
| Degenerate Table (ad + bc = 0) | Undefined | "Degenerate Zero-Cross-Product Table" | Guard |
Yule’s Q Coefficient of Association: (a*d - b*c) / (a*d + b*c)
Computes Yule’s Q across 2×2 cell counts a (B2), b (C2), c (B3), and d (C3).
=IF(OR(MIN(B2:C3) < 0, (B2 * C3 + C2 * B3) = 0), "Invalid or Degenerate 2x2 Table", ROUND(((B2 * C3) - (C2 * B3)) / ((B2 * C3) + (C2 * B3)), 4))Yule’s Y Coefficient of Colligation: (SQRT(ad) - SQRT(bc)) / (SQRT(ad) + SQRT(bc))
Evaluates the square-root-compressed colligation index Y on [-1, +1].
=IF(OR(MIN(B2:C3) < 0, (B2 * C3 + C2 * B3) = 0), "Invalid or Degenerate 2x2 Table", ROUND((SQRT(B2 * C3) - SQRT(C2 * B3)) / (SQRT(B2 * C3) + SQRT(C2 * B3)), 4))Try the example
Use the original values or change them to test your understanding.